Answer
Yes, see explanations.
Work Step by Step
Step 1. Given $y=sin(x-sin(x))$, we have $y'=cos(x-sin(x))(1-cos(x))$ which gives the slope of tangent lines to the curve.
Step 2. For horizontal tangent lines, let $y'=0$; we have $cos(x-sin(x))(1-cos(x))=0$ and the possible solutions are $cos(x-sin(x))=0$ and $(1-cos(x))=0$
Step 3. Case 1: $x-sin(x)=k\pi+\pi/2$, there are infinite solutions and three examples are given in the figure. These solutions correspond to the extrema of the function.
Step 4. Case 2: $cos(x)=1$ and $x=2k\pi$ where $k$ is an integer. There solutions are all on the x-axis.
Step 5. Within the domain of $[-2\pi, 2\pi]$ given in the exercise, the solutions are $x=0,\pm2\pi,\pm2.31,\pm3.97$. These 7 locations can also be seen from the original figure of the Exercise.
